You should avoid using the column name 'date' and any other reserved words as listed here: https://dev.mysql.com/doc/refman/5.5/en/keywords.html
Your select list for date is named "select", but your code is referencing the field as "date".... They have to match, so line:
$WADbSearch1->addComparisonFromList("date","date","AND","=",0);
should be:
$WADbSearch1->addComparisonFromList("date","select","AND","=",0);
or you need to update the select name to "date" to match the php code.
Your checkbox names are "CheckboxGroup1". If you want your users to be able to select more than one, they would have to be named: "CheckboxGroup1[]". If they can only select one then you should be using radio buttons instead.
and then you have a similar issue with the column name reference for that search as well. This code:
$WADbSearch1->addComparisonFromCheck("category","category","".$row_WADAnews['category'] ."","","AND","=","=",0);
should be:
$WADbSearch1->addComparisonFromList("category","CheckboxGroup1","AND","=",0);
Groups of checkboxes act just like multi-select lists.